Best Known (90−20, 90, s)-Nets in Base 2
(90−20, 90, 138)-Net over F2 — Constructive and digital
Digital (70, 90, 138)-net over F2, using
- trace code for nets [i] based on digital (10, 30, 46)-net over F8, using
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F8 with g(F) = 9, N(F) = 45, and 1 place with degree 2 [i] based on function field F/F8 with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (10, 45)-sequence over F8, using
(90−20, 90, 236)-Net over F2 — Digital
Digital (70, 90, 236)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(290, 236, F2, 2, 20) (dual of [(236, 2), 382, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(290, 256, F2, 2, 20) (dual of [(256, 2), 422, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(290, 512, F2, 20) (dual of [512, 422, 21]-code), using
- 1 times truncation [i] based on linear OA(291, 513, F2, 21) (dual of [513, 422, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 513 | 218−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(291, 513, F2, 21) (dual of [513, 422, 22]-code), using
- OOA 2-folding [i] based on linear OA(290, 512, F2, 20) (dual of [512, 422, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(290, 256, F2, 2, 20) (dual of [(256, 2), 422, 21]-NRT-code), using
(90−20, 90, 2304)-Net in Base 2 — Upper bound on s
There is no (70, 90, 2305)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1241 952547 721257 670295 761664 > 290 [i]